Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Abstract

Let X ^*_n1 , ... X ^*_nn be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M ^*_n = \max\{X ^*_n1 , ... X ^*_nn }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j ∈ N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M ^*_n and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let α _n, β_n ∈ N_n, $M(A_n ) = \max \{ X_{n1} ,...,X_{n\alpha _n } \}$, $M(B_n ) = \max \{ X_{n,n - \beta _n + 1} ,...,X_{nn} \}$ and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case α _n → α, β _n/α_n → γ > 0, as n → ∞.